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A symplectic test of the Lfunctions ratios conjecture
 Int. Math. Res. Notices, 2008, article ID rnm
"... ABSTRACT. Recently Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] conjectured formulas for the averages over a family of ratios of products of shifted Lfunctions. Their Lfunctions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from nlevel correlations and den ..."
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ABSTRACT. Recently Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] conjectured formulas for the averages over a family of ratios of products of shifted Lfunctions. Their Lfunctions Ratios Conjecture predicts both the main and lower order terms for many problems, ranging from nlevel correlations and densities to mollifiers and moments to vanishing at the central point. There are now many results showing agreement between the main terms of number theory and random matrix theory; however, there are very few families where the lower order terms are known. These terms often depend on subtle arithmetic properties of the family, and provide a way to break the universality of behavior. The Lfunctions Ratios Conjecture provides a powerful and tractable way to predict these terms. We test a specific case here, that of the 1level density for the symplectic family of quadratic Dirichlet characters arising from even fundamental discriminants d ≤ X. For test functions supported in (−1/3, 1/3) we calculate all the lower order terms up to size O(X −1/2+ǫ) and observe perfect agreement with the conjecture (for test functions supported in (−1, 1) we show agreement up to errors of size O(X −ǫ) for any ǫ). Thus for this family and suitably restricted test functions, we completely verify the Ratios Conjecture’s prediction for the 1level density. 1.
A RANDOM MATRIX MODEL FOR ELLIPTIC CURVE LFUNCTIONS OF FINITE CONDUCTOR
, 2011
"... Abstract. We propose a random matrix model for families of elliptic curve Lfunctions of finite conductor. A repulsion of the critical zeros of these Lfunctions away from the center of the critical strip was observed numerically by S. J. Miller in 2006 [50]; such behaviour deviates qualitatively fr ..."
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Abstract. We propose a random matrix model for families of elliptic curve Lfunctions of finite conductor. A repulsion of the critical zeros of these Lfunctions away from the center of the critical strip was observed numerically by S. J. Miller in 2006 [50]; such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros (for large conductors this distribution is expected to approach the onelevel density of eigenvalues of orthogonal matrices after appropriate rescaling). Our purpose here is to provide a random matrix model for Miller’s surprising discovery. We consider the family of even quadratic twists of a given elliptic curve. The main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra. We call this subensemble of SO(2N) the excised orthogonal ensemble. The sievingoff of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of Lfunctions implied by the formulæ of Waldspurger and KohnenZagier. The cutoff scale